Handout 2: Symmetry in Crystallography

Chem 4850/6850/8850X-ray Crystallography

The University of Toledo

cora.lind@utoledo.edu

Symmetry and crystals

Imagine…

- having to describe an infinite crystal with an infinite number of atoms

- or even a finite crystal, with some 1020 atoms

Sounds horrible? Well, there’s symmetry to help you out! Instead

of an infinite number of atoms, you only need to describe the

contents of one unit cell, the structural repeating motif…

- and life could be even easier, if there are symmetry elements present

inside the unit cell!

- you only need to describe the asymmetric unit if this is the case

Lattice symmetry

Lattice symmetry

refers to the unit cell

size and shape

Without rules, there

would be an infinite

number of different

unit cells to describe

any given lattice

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Unit cell choice

By convention, a unit cell is chosen as

- the smallest possible repeat unit

- which has the highest symmetry

This can result in primitive unit cells or centered unit cells

- not all crystal systems can be centered by this definition

- the seven crystal systems in combination with the centering operations

give rise to the 14 Bravais lattices

An example of unit cell choice

According to our definitions, the centered cell would be preferred

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

The 14 Bravais lattices

“Elements of X-ray Diffraction”, Cullity and Stock, Prentice Hall College Div., 3rd edition, 2001.

Why can only some lattices be centered?

a’

a’

a a.

.

.

.

A tetragonal base

centered cell can

always be transformed

into a tetragonal

primitive cell

Why can only some lattices be centered? (2)

An orthorhombic base centered cell cannot be

transformed into a primitive cell without losing the

orthorhombic symmetry

a

bca’

a’c

..

90

Symmetry elements

When talking about symmetry operations, we must distinguish

- point symmetry elements

- translational symmetry

Point symmetry elements will always leave at least one point

unchanged

- rotation axes

- mirror planes

- rotation-inversion axes

Rotation axes

Example: A two-fold

rotation axis

- no change in handedness

- referred to as “proper

symmetry operation”

An n-fold rotation axis

will rotate the object by

360/n°

Symbol: n (e.g., 2, 3, 4, 6)

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Mirror planes

A mirror plane changes

the handedness of the

object it is operating on

- cannot exist in crystals of

an enantiomerically pure

substance

- referred to as “improper

symmetry operation”

Symbol: m

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Inversion centers

“Turning an object inside

out”

Equivalent to a “point

reflection” through the

inversion center

- similar to focal point of a

lense

- changes handedness

Symbol: i

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Rotation-inversion centers

Rotation followed by

inversion

An inversion center can

be regarded as a “one-fold

rotation” followed by an

inversion

Symbol: -n or n

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Watch out…

Crystallographers work with rotation-inversion axes

If you take a class in Group Theory or another subject

involving symmetry operations, your teacher may not

consider rotation-inversion axes a symmetry operation

- they use rotation-reflection axes (symbol: Sn)

- rotation-inversion and rotation-reflection axes are NOT the same!

- an inversion center corresponds to a “–1 axis”, but to an S2 axis!

- however, any compound that has a –4 axis will also possess an S4 axis

S4 versus -4

Combining symmetry operations

An object can possess several symmetry elements

Not all symmetry elements can be combined arbitrarily

- for example, two perpendicular two-fold axes imply the existence of a

third perpendicular two-fold

Translational symmetry in 3D imposes limitations

- only 2, 3, 4 and 6-fold rotation axes allow for space filling

translational symmetry

The allowed combinations of symmetry elements are called

point groups

- there are 32 point groups that give rise to periodicity in 3D

Space filling repeat patterns

Only 2, 3, 4 and 6-fold rotations

can produce space filling patterns

“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994.

Point groups

Show 3D repeat

pattern

Contain symmetry

elements

32 point groups exist

“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994.

Space groups

When talking about crystal structures, people will usually

report the space group of a crystal

Space groups are made up from

- lattice symmetry (translational)

- point symmetry (not translational)

- glide and/or screw axes (some translational component)

There are 230 space groups

7 crystal systems

14 Bravais lattices

32 point groups

230 space groups

Screw axes

A 21 screw axis translates an object by half a unit cell in the

direction of the screw axis, followed by a 180° rotation

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Higher order screws

A Cn screw axis translates an object by the unit cell dimension

multiplied by n/C along the direction of the screw axis, followed by

a C-fold rotation

“Structure Determination by X-ray Crystallography”, Ladd and Palmer, Plenum, 1994.

Glide planes

A glide plane corresponds to a reflection-translation operation

- reflection through the glide plane

- translation within/parallel to the glide plane

- exact translation depends on type of glide

There are a, b, c, n and d glide planes

- a, b and c glides correspond to translations of ½ a, ½ b and ½ c

respectively

- called “axial glide planes”

- n glide corresponds to a translation of ½ a + ½ b, ½ a + ½ c, or ½ b + ½ c

- called “diagonal glide plane”

- d glide corresponds to a translation of ¼ a + ¼ b, ¼ a + ¼ c, or ¼ b + ¼ c

- called “diamond glide plane”

Example of an a glide plane

“Crystal Structure Analysis for Chemists and Biologists”, Glusker, Lewis and Rossi, VCH, 1994.

Limitations on combination of symmetry elements

As for point groups, not all symmetry elements can be combined

arbitrarily

For three dimensional lattices

- 14 Bravais lattices

- 32 point groups

- but only 230 space groups

For two dimensional lattices

- 5 lattices

- 10 point groups

- but only 17 plane groups

Graphical symbols used for symmetry operations“International Tables for Crystallography, Vol. A”, Kluwer, 1993.

Graphical symbols (2)“International Tables for Crystallography, Vol. A”, Kluwer, 1993.

Symmetry elements parallel to the plane of projection“International Tables for Crystallography, Vol. A”, Kluwer, 1993.

Symmetry elements inclined to the plane of projection“International Tables for Crystallography, Vol. A”, Kluwer, 1993.